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   "metadata": {},
   "source": [
    "<img src=\"http://hilpisch.com/tpq_logo.png\" alt=\"The Python Quants\" width=\"35%\" align=\"right\" border=\"0\"><br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Python for Finance (2nd ed.)\n",
    "\n",
    "**Mastering Data-Driven Finance**\n",
    "\n",
    "&copy; Dr. Yves J. Hilpisch | The Python Quants GmbH\n",
    "\n",
    "<img src=\"http://hilpisch.com/images/py4fi_2nd_shadow.png\" width=\"300px\" align=\"left\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mathematical Tools"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Approximation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from pylab import plt, mpl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.style.use('seaborn-v0_8')\n",
    "mpl.rcParams['font.family'] = 'serif'\n",
    "%config InlineBackend.figure_format = 'svg'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "uuid": "2326c3ad-f244-4f48-8b68-851bd2347d57"
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return np.sin(x) + 0.5 * x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_plot(x, y, styles, labels, axlabels):\n",
    "    plt.figure(figsize=(10, 6))\n",
    "    for i in range(len(x)):\n",
    "        plt.plot(x[i], y[i], styles[i], label=labels[i])\n",
    "        plt.xlabel(axlabels[0])\n",
    "        plt.ylabel(axlabels[1])\n",
    "    plt.legend(loc=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "uuid": "c09f73d2-c2a5-4c6d-a2f1-08a191378417"
   },
   "outputs": [],
   "source": [
    "x = np.linspace(-2 * np.pi, 2 * np.pi, 50)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "uuid": "96d2bd1b-8883-486d-920d-b610aeb076a8"
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   "outputs": [
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Monomials as Basis Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "res = np.polyfit(x, f(x), deg=1, full=True)  "
   ]
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   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 4.28841952e-01, -4.31775426e-17]),\n",
       " array([21.03238686]),\n",
       " 2,\n",
       " array([1., 1.]),\n",
       " 1.1102230246251565e-14)"
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   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
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   "source": [
    "ry = np.polyval(res[0], x)  "
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   "execution_count": 10,
   "metadata": {
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   "source": [
    "create_plot([x, x], [f(x), ry], ['b', 'r.'],\n",
    "            ['f(x)', 'regression'], ['x', 'f(x)'])\n",
    "# plt.savefig('../../images/ch11/math_02.png');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "uuid": "096bb07a-55f7-45de-8734-2a76d8749d53"
   },
   "outputs": [],
   "source": [
    "reg = np.polyfit(x, f(x), deg=5)\n",
    "ry = np.polyval(reg, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "uuid": "5e17309e-e8e2-4df9-b841-0f57d983b89e"
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   "source": [
    "create_plot([x, x], [f(x), ry], ['b', 'r.'],\n",
    "            ['f(x)', 'regression'], ['x', 'f(x)'])"
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    "reg = np.polyfit(x, f(x), 7)\n",
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   "execution_count": 14,
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   "execution_count": 16,
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   "source": [
    "create_plot([x, x], [f(x), ry], ['b', 'r.'],\n",
    "            ['f(x)', 'regression'], ['x', 'f(x)'])"
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   "metadata": {},
   "source": [
    "#### Individual Basis Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "uuid": "b4f05890-56e0-4f29-9d61-bd9948ad8af0"
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   "outputs": [],
   "source": [
    "matrix = np.zeros((3 + 1, len(x)))  \n",
    "matrix[3, :] = x ** 3  \n",
    "matrix[2, :] = x ** 2  \n",
    "matrix[1, :] = x  \n",
    "matrix[0, :] = 1  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "uuid": "c8963eee-4bc8-4ef2-a172-d4b64fd065a3"
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   "source": [
    "reg = np.linalg.lstsq(matrix.T, f(x), rcond=None)[0]  "
   ]
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  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "uuid": "efd077d1-9c8a-4961-be95-400f83cd679e"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.    ,  0.5628, -0.    , -0.0054])"
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     },
     "execution_count": 19,
     "metadata": {},
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   "source": [
    "reg.round(4)  "
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  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "uuid": "efb7b252-d0f8-4263-b2be-4d9588ab06a7"
   },
   "outputs": [],
   "source": [
    "ry = np.dot(reg, matrix)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "uuid": "1b1953fe-83a2-436b-8cd4-69c5abf6d2e1"
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   "source": [
    "create_plot([x, x], [f(x), ry], ['b', 'r.'],\n",
    "            ['f(x)', 'regression'], ['x', 'f(x)'])"
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   "execution_count": 22,
   "metadata": {},
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    "matrix[3, :] = np.sin(x)  "
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    "reg = np.linalg.lstsq(matrix.T, f(x), rcond=None)[0]"
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   "execution_count": 24,
   "metadata": {
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    {
     "data": {
      "text/plain": [
       "array([0. , 0.5, 0. , 1. ])"
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     "execution_count": 24,
     "metadata": {},
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   "source": [
    "ry = np.dot(reg, matrix)"
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   "metadata": {
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    {
     "data": {
      "text/plain": [
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     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
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   "execution_count": 27,
   "metadata": {
    "uuid": "6bf80137-3a52-483b-a557-b092bbf23b36"
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    {
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      "text/plain": [
       "2.099263639382087e-31"
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     },
     "execution_count": 27,
     "metadata": {},
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   ],
   "source": [
    "np.mean((f(x) - ry) ** 2)  "
   ]
  },
  {
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   "execution_count": 28,
   "metadata": {
    "uuid": "58d9db31-5885-4fba-8ae7-2e962a0963ca"
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      "text/plain": [
       "<Figure size 1000x600 with 1 Axes>"
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "create_plot([x, x], [f(x), ry], ['b', 'r.'],\n",
    "            ['f(x)', 'regression'], ['x', 'f(x)'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Noisy Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "uuid": "75d3a6a6-a940-4a49-b35d-29f21880ab95"
   },
   "outputs": [],
   "source": [
    "xn = np.linspace(-2 * np.pi, 2 * np.pi, 50)  \n",
    "xn = xn + 0.15 * np.random.standard_normal(len(xn))  \n",
    "yn = f(xn) + 0.25 * np.random.standard_normal(len(xn))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "uuid": "f6f9c05f-1f96-48ee-aaca-f4d80c3d3ac5"
   },
   "outputs": [],
   "source": [
    "reg = np.polyfit(xn, yn, 7)\n",
    "ry = np.polyval(reg, xn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "uuid": "9a475222-3bfd-4300-951b-94e60792c6da"
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      "text/plain": [
       "<Figure size 1000x600 with 1 Axes>"
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "create_plot([x, x], [f(x), ry], ['b', 'r.'],\n",
    "            ['f(x)', 'regression'], ['x', 'f(x)'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Unsorted Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "uuid": "8ea85cdb-47f2-4967-b684-7894d9964e76"
   },
   "outputs": [],
   "source": [
    "xu = np.random.rand(50) * 4 * np.pi - 2 * np.pi  \n",
    "yu = f(xu)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "uuid": "0034edf5-1cef-4eea-be44-c69103fe6eb2"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.48  1.37 -4.14 -2.28 -0.31 -2.64 -0.9  -4.93 -3.1  -2.55]\n",
      "[ 1.74  1.67 -1.23 -1.9  -0.46 -1.8  -1.23 -1.49 -1.59 -1.83]\n"
     ]
    }
   ],
   "source": [
    "print(xu[:10].round(2))  \n",
    "print(yu[:10].round(2))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "uuid": "d7f5f003-1cb8-4432-a8d6-cb4bef1a101a"
   },
   "outputs": [],
   "source": [
    "reg = np.polyfit(xu, yu, 5)\n",
    "ry = np.polyval(reg, xu)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "uuid": "40177962-0363-479c-bdbd-451a4c043060"
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      "text/plain": [
       "<Figure size 1000x600 with 1 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "create_plot([xu, xu], [yu, ry], ['b.', 'ro'],\n",
    "            ['f(x)', 'regression'], ['x', 'f(x)'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Multiple Dimensions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "uuid": "82b95a7b-9e3e-4dc8-b313-1af775b06b8b"
   },
   "outputs": [],
   "source": [
    "def fm(p):\n",
    "    x, y = p\n",
    "    return np.sin(x) + 0.25 * x + np.sqrt(y) + 0.05 * y ** 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(0, 10, 20)\n",
    "y = np.linspace(0, 10, 20)\n",
    "X, Y = np.meshgrid(x, y)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "uuid": "b03b67ac-f7df-41d1-9dab-1e074e8738fa"
   },
   "outputs": [],
   "source": [
    "Z = fm((X, Y))\n",
    "x = X.flatten()  \n",
    "y = Y.flatten()  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "uuid": "52a91ef7-33c4-4de1-b69b-ea4d740aa252"
   },
   "outputs": [],
   "source": [
    "from mpl_toolkits.mplot3d import Axes3D  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "uuid": "52a91ef7-33c4-4de1-b69b-ea4d740aa252"
   },
   "outputs": [
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   ],
   "source": [
    "fig = plt.figure(figsize=(10, 6))\n",
    "ax = fig.add_subplot(projection='3d')\n",
    "surf = ax.plot_surface(X, Y, Z, rstride=2, cstride=2,\n",
    "                       cmap='coolwarm', linewidth=0.5,\n",
    "                       antialiased=True)\n",
    "ax.set_xlabel('x')\n",
    "ax.set_ylabel('y')\n",
    "ax.set_zlabel('f(x, y)')\n",
    "fig.colorbar(surf, shrink=0.5, aspect=5);"
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  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "uuid": "5918f2cf-3ead-4b80-980e-4a375ee159db"
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   "outputs": [],
   "source": [
    "matrix = np.zeros((len(x), 6 + 1))\n",
    "matrix[:, 6] = np.sqrt(y)  \n",
    "matrix[:, 5] = np.sin(x)  \n",
    "matrix[:, 4] = y ** 2\n",
    "matrix[:, 3] = x ** 2\n",
    "matrix[:, 2] = y\n",
    "matrix[:, 1] = x\n",
    "matrix[:, 0] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "uuid": "b9eb74bd-9280-4d8b-ae7d-8853911389cb"
   },
   "outputs": [],
   "source": [
    "reg = np.linalg.lstsq(matrix, fm((x, y)), rcond=None)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "RZ = np.dot(matrix, reg).reshape((20, 20))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 1000x600 with 2 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(subplot_kw={\"projection\": \"3d\"}, figsize=(10, 6))\n",
    "surf1 = ax.plot_surface(X, Y, Z, rstride=2, cstride=2,\n",
    "            cmap=mpl.cm.coolwarm, linewidth=0.5,\n",
    "            antialiased=True)  \n",
    "surf2 = ax.plot_wireframe(X, Y, RZ, rstride=2, cstride=2,\n",
    "                          label='regression')  \n",
    "ax.set_xlabel('x')\n",
    "ax.set_ylabel('y')\n",
    "ax.set_zlabel('f(x, y)')\n",
    "ax.legend()\n",
    "fig.colorbar(surf, shrink=0.5, aspect=5, ax=ax);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Interpolation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "uuid": "5e345542-f425-4630-9eae-0a446927ad73"
   },
   "outputs": [],
   "source": [
    "import scipy.interpolate as spi  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "uuid": "20200433-3e90-4bb6-be5a-8e6a1360989e"
   },
   "outputs": [],
   "source": [
    "x = np.linspace(-2 * np.pi, 2 * np.pi, 25)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "uuid": "6e3caa90-2a2a-4429-8536-ab25091a5640"
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return np.sin(x) + 0.5 * x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "uuid": "afdb9c45-1b12-4358-ba3e-a1bc3c4d32db"
   },
   "outputs": [],
   "source": [
    "ipo = spi.splrep(x, f(x), k=1)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "uuid": "7217594a-3398-4dfd-9d16-f90ee0f24bd9"
   },
   "outputs": [],
   "source": [
    "iy = spi.splev(x, ipo)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "uuid": "b9cacc73-786b-4c6b-ad5f-e4d22729c17e"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(f(x), iy)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "uuid": "89d96d0e-dabd-490f-acad-15bba904b546"
   },
   "outputs": [
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   "source": [
    "create_plot([x, x], [f(x), iy], ['b', 'ro'],\n",
    "            ['f(x)', 'interpolation'], ['x', 'f(x)'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "uuid": "3daf1f42-54d8-4150-aab6-5188d1c23bdb"
   },
   "outputs": [],
   "source": [
    "xd = np.linspace(1.0, 3.0, 50)  \n",
    "iyd = spi.splev(xd, ipo)"
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  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
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   "source": [
    "create_plot([xd, xd], [f(xd), iyd], ['b', 'ro'],\n",
    "            ['f(x)', 'interpolation'], ['x', 'f(x)'])"
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  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "uuid": "c3d9f85f-671f-4353-beab-fad9db9f8aa4"
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   "outputs": [],
   "source": [
    "ipo = spi.splrep(x, f(x), k=3)  \n",
    "iyd = spi.splev(xd, ipo)  "
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  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "uuid": "8c67b514-160e-40e3-8f69-da81d7579b2d"
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   "execution_count": 56,
   "metadata": {
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   "outputs": [
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     "execution_count": 56,
     "metadata": {},
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   ],
   "source": [
    "np.mean((f(xd) - iyd) ** 2)  "
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   "execution_count": 57,
   "metadata": {
    "uuid": "0c3e96f1-8732-4ce6-a6ed-bbaae732d950"
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   ],
   "source": [
    "create_plot([xd, xd], [f(xd), iyd], ['b', 'ro'],\n",
    "            ['f(x)', 'interpolation'], ['x', 'f(x)'])"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Convex Optimization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "uuid": "a09a918e-760c-4e9c-95b8-fe3b568dacdb"
   },
   "outputs": [],
   "source": [
    "def fm(p):\n",
    "    x, y = p\n",
    "    return (np.sin(x) + 0.05 * x ** 2\n",
    "          + np.sin(y) + 0.05 * y ** 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "uuid": "1f4a147b-d9d2-42d5-8553-dd95d75d8251"
   },
   "outputs": [],
   "source": [
    "x = np.linspace(-10, 10, 50)\n",
    "y = np.linspace(-10, 10, 50)\n",
    "X, Y = np.meshgrid(x, y)\n",
    "Z = fm((X, Y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "uuid": "b3a1d3b7-25eb-4ac8-81ed-ad740fd1a297"
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       "<Figure size 1000x600 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(subplot_kw={\"projection\": \"3d\"}, figsize=(10, 6))\n",
    "surf = ax.plot_surface(X, Y, Z, rstride=2, cstride=2,\n",
    "                       cmap='coolwarm', linewidth=0.5,\n",
    "                       antialiased=True)\n",
    "ax.set_xlabel('x')\n",
    "ax.set_ylabel('y')\n",
    "ax.set_zlabel('f(x, y)')\n",
    "fig.colorbar(surf, shrink=0.5, aspect=5);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Global Optimization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "uuid": "72b2e44d-2127-412e-9046-2b0c7c5bf7c6"
   },
   "outputs": [],
   "source": [
    "import scipy.optimize as sco  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "uuid": "e6c829bb-f358-430d-9e8c-781e9d15c9e6"
   },
   "outputs": [],
   "source": [
    "def fo(p):\n",
    "    x, y = p\n",
    "    z = np.sin(x) + 0.05 * x ** 2 + np.sin(y) + 0.05 * y ** 2\n",
    "    if output == True:\n",
    "        print('%8.4f | %8.4f | %8.4f' % (x, y, z))  \n",
    "    return z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "uuid": "d1f54d32-6cb5-4cd2-8a95-ea3b92c87266"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-10.0000 | -10.0000 |  11.0880\n",
      "-10.0000 |  -5.0000 |   7.7529\n",
      "-10.0000 |   0.0000 |   5.5440\n",
      "-10.0000 |   5.0000 |   5.8351\n",
      "-10.0000 |  10.0000 |  10.0000\n",
      " -5.0000 | -10.0000 |   7.7529\n",
      " -5.0000 |  -5.0000 |   4.4178\n",
      " -5.0000 |   0.0000 |   2.2089\n",
      " -5.0000 |   5.0000 |   2.5000\n",
      " -5.0000 |  10.0000 |   6.6649\n",
      "  0.0000 | -10.0000 |   5.5440\n",
      "  0.0000 |  -5.0000 |   2.2089\n",
      "  0.0000 |   0.0000 |   0.0000\n",
      "  0.0000 |   5.0000 |   0.2911\n",
      "  0.0000 |  10.0000 |   4.4560\n",
      "  5.0000 | -10.0000 |   5.8351\n",
      "  5.0000 |  -5.0000 |   2.5000\n",
      "  5.0000 |   0.0000 |   0.2911\n",
      "  5.0000 |   5.0000 |   0.5822\n",
      "  5.0000 |  10.0000 |   4.7471\n",
      " 10.0000 | -10.0000 |  10.0000\n",
      " 10.0000 |  -5.0000 |   6.6649\n",
      " 10.0000 |   0.0000 |   4.4560\n",
      " 10.0000 |   5.0000 |   4.7471\n",
      " 10.0000 |  10.0000 |   8.9120\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([0., 0.])"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "output = True\n",
    "sco.brute(fo, ((-10, 10.1, 5), (-10, 10.1, 5)), finish=None)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "output = False\n",
    "opt1 = sco.brute(fo, ((-10, 10.1, 0.1), (-10, 10.1, 0.1)), finish=None)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "uuid": "dc42167b-4e8f-462c-8aad-a94afdf0a8bc"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.4, -1.4])"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "opt1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "uuid": "ec2669cd-36a5-4b79-a90c-761a39b12044"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.7748994599769203"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fm(opt1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Local Optimization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " -1.4000 |  -1.4000 |  -1.7749\n",
      " -1.4700 |  -1.4000 |  -1.7743\n",
      " -1.4000 |  -1.4700 |  -1.7743\n",
      " -1.3300 |  -1.4700 |  -1.7696\n",
      " -1.4350 |  -1.4175 |  -1.7756\n",
      " -1.4350 |  -1.3475 |  -1.7722\n",
      " -1.4088 |  -1.4394 |  -1.7755\n",
      " -1.4438 |  -1.4569 |  -1.7751\n",
      " -1.4328 |  -1.4427 |  -1.7756\n",
      " -1.4591 |  -1.4208 |  -1.7752\n",
      " -1.4213 |  -1.4347 |  -1.7757\n",
      " -1.4235 |  -1.4096 |  -1.7755\n",
      " -1.4305 |  -1.4344 |  -1.7757\n",
      " -1.4168 |  -1.4516 |  -1.7753\n",
      " -1.4305 |  -1.4260 |  -1.7757\n",
      " -1.4396 |  -1.4257 |  -1.7756\n",
      " -1.4259 |  -1.4325 |  -1.7757\n",
      " -1.4259 |  -1.4241 |  -1.7757\n",
      " -1.4304 |  -1.4177 |  -1.7757\n",
      " -1.4270 |  -1.4288 |  -1.7757\n",
      " -1.4316 |  -1.4307 |  -1.7757\n",
      " -1.4273 |  -1.4258 |  -1.7757\n",
      " -1.4239 |  -1.4285 |  -1.7757\n",
      " -1.4288 |  -1.4266 |  -1.7757\n",
      " -1.4285 |  -1.4297 |  -1.7757\n",
      " -1.4276 |  -1.4267 |  -1.7757\n",
      " -1.4258 |  -1.4289 |  -1.7757\n",
      " -1.4281 |  -1.4272 |  -1.7757\n",
      " -1.4286 |  -1.4252 |  -1.7757\n",
      " -1.4274 |  -1.4279 |  -1.7757\n",
      " -1.4279 |  -1.4283 |  -1.7757\n",
      " -1.4277 |  -1.4271 |  -1.7757\n",
      "Optimization terminated successfully.\n",
      "         Current function value: -1.775726\n",
      "         Iterations: 16\n",
      "         Function evaluations: 32\n"
     ]
    }
   ],
   "source": [
    "output = True\n",
    "opt2 = sco.fmin(fo, opt1, xtol=0.001, ftol=0.001,\n",
    "                maxiter=25, maxfun=200)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "uuid": "e451a278-e2ae-4d86-b357-f230468052a4"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.42743423, -1.42786531])"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "opt2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "scrolled": true,
    "uuid": "0352d556-6ea7-47a6-9c76-e4a56c0efed7"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.7757255920546628"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fm(opt2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "uuid": "ff9beaf9-73d7-4157-85cb-24783abae775"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.015826\n",
      "         Iterations: 46\n",
      "         Function evaluations: 86\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([4.2710728 , 4.27106945])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "output = False\n",
    "sco.fmin(fo, (2.0, 2.0), maxiter=250)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Constrained Optimization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "uuid": "703a8ac0-1c74-4851-9fd3-7ac404ee3446"
   },
   "outputs": [],
   "source": [
    "import math"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "uuid": "703a8ac0-1c74-4851-9fd3-7ac404ee3446"
   },
   "outputs": [],
   "source": [
    "def Eu(p):  \n",
    "    s, b = p\n",
    "    return -(0.5 * math.sqrt(s * 15 + b * 5) +\n",
    "             0.5 * math.sqrt(s * 5 + b * 12))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "uuid": "703a8ac0-1c74-4851-9fd3-7ac404ee3446"
   },
   "outputs": [],
   "source": [
    "cons = ({'type': 'ineq',\n",
    "         'fun': lambda p: 100 - p[0] * 10 - p[1] * 10})  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "uuid": "703a8ac0-1c74-4851-9fd3-7ac404ee3446"
   },
   "outputs": [],
   "source": [
    "bnds = ((0, 1000), (0, 1000))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "uuid": "8f989832-637a-4188-b096-26ceeabd77ea"
   },
   "outputs": [],
   "source": [
    "result = sco.minimize(Eu, [5, 5], method='SLSQP',\n",
    "                       bounds=bnds, constraints=cons)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "uuid": "67c11995-124c-4f9d-8edf-1364b412e754"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       " message: Optimization terminated successfully\n",
       " success: True\n",
       "  status: 0\n",
       "     fun: -9.700883611487832\n",
       "       x: [ 8.025e+00  1.975e+00]\n",
       "     nit: 5\n",
       "     jac: [-4.851e-01 -4.849e-01]\n",
       "    nfev: 16\n",
       "    njev: 5"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "uuid": "53899a89-8532-4f22-a17f-54d106412867"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([8.02547122, 1.97452878])"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "result['x']  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "uuid": "b8548809-fe3a-4fa4-9496-8d3b6a177973"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.700883611487832"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-result['fun']  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "uuid": "7453e87b-a70d-47b9-8a12-c5c0f01e4b86"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100.0"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(result['x'], [10, 10])  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "uuid": "6a38567d-4a2d-464f-8de3-7687e1d2916d"
   },
   "outputs": [],
   "source": [
    "import scipy.integrate as sci"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "uuid": "0954e5b6-9874-4bad-a2fb-28273a015e4a"
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return np.sin(x) + 0.5 * x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "uuid": "afb6151f-5743-4950-980e-18047e066ebc"
   },
   "outputs": [],
   "source": [
    "x = np.linspace(0, 10)\n",
    "y = f(x)\n",
    "a = 0.5  \n",
    "b = 9.5  \n",
    "Ix = np.linspace(a, b)  \n",
    "Iy = f(Ix)  #  <4>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "uuid": "ad78923f-e4c1-4381-9a24-33b4f1ea3c56"
   },
   "outputs": [],
   "source": [
    "from matplotlib.patches import Polygon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "uuid": "ad78923f-e4c1-4381-9a24-33b4f1ea3c56"
   },
   "outputs": [
    {
     "data": {
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       "    <dc:date>2023-10-15T07:56:24.133932</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
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       "  </rdf:RDF>\n",
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       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
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       "L 614.4 380.478125 \n",
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       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 1000x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(10, 6))\n",
    "plt.plot(x, y, 'b', linewidth=2)\n",
    "plt.ylim(bottom=0)\n",
    "Ix = np.linspace(a, b)\n",
    "Iy = f(Ix)\n",
    "verts = [(a, 0)] + list(zip(Ix, Iy)) + [(b, 0)]\n",
    "poly = Polygon(verts, facecolor='0.7', edgecolor='0.5')\n",
    "ax.add_patch(poly)\n",
    "plt.text(0.75 * (a + b), 1.5, r\"$\\int_a^b f(x)dx$\",\n",
    "         horizontalalignment='center', fontsize=20)\n",
    "plt.figtext(0.9, 0.075, '$x$')\n",
    "plt.figtext(0.075, 0.9, '$f(x)$')\n",
    "ax.set_xticks((a, b))\n",
    "ax.set_xticklabels(('$a$', '$b$'))\n",
    "ax.set_yticks([f(a), f(b)]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Numerical Integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "uuid": "6d3aa9c8-58c0-46a3-a89f-ca471a1f83e4"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "24.366995967084605"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sci.fixed_quad(f, a, b)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "uuid": "0ddebba6-5357-4ce9-b829-d752132d12df"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "24.374754718086752"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sci.quad(f, a, b)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "uuid": "3b1779d6-c831-449d-964f-e2cc869847e7"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "24.374754718086713"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sci.romberg(f, a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "uuid": "5499809d-f2b1-4cc9-8808-511f4e82c5ed"
   },
   "outputs": [],
   "source": [
    "xi = np.linspace(0.5, 9.5, 25)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "uuid": "aec239bc-cdd9-4477-b47c-4d0136ed686d"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "24.352733271544516"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sci.trapz(f(xi), xi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "uuid": "ce33e420-5143-4d54-b23f-0040f54723fc"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "24.37496418455075"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sci.simps(f(xi), xi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integration by Simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "uuid": "bb840eb2-0e13-40eb-aea7-0f106dcf9f41"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "24.804762279331463\n",
      "26.522918898332378\n",
      "26.265547519223976\n",
      "26.02770339943824\n",
      "24.99954181440844\n",
      "23.881810141621663\n",
      "23.527912274843253\n",
      "23.507857658961207\n",
      "23.67236746066989\n",
      "23.679410416062886\n",
      "24.424401707879305\n",
      "24.239005346819056\n",
      "24.115396924962802\n",
      "24.424191987566726\n",
      "23.924933080533783\n",
      "24.19484212027875\n",
      "24.117348378249833\n",
      "24.100690929662274\n",
      "23.76905109847816\n"
     ]
    }
   ],
   "source": [
    "for i in range(1, 20):\n",
    "    np.random.seed(1000)\n",
    "    x = np.random.random(i * 10) * (b - a) + a  \n",
    "    print(np.mean(f(x)) * (b - a))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symbolic Computation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "uuid": "f01d358d-8094-426c-8e69-657218d8c319"
   },
   "outputs": [],
   "source": [
    "import sympy as sy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "uuid": "2cb7d5d8-ae45-4001-863e-714df216a656"
   },
   "outputs": [],
   "source": [
    "x = sy.Symbol('x')  \n",
    "y = sy.Symbol('y')  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "uuid": "7bf55a76-f348-4820-af94-a83ed233d9a2"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.symbol.Symbol"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "uuid": "a8f0388b-1df6-4389-877c-af5d9f141708"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sqrt{x}$"
      ],
      "text/plain": [
       "sqrt(x)"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.sqrt(x)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "uuid": "f308fe2e-a5bc-4c88-a7f6-a4f09a568f2f"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sqrt{x} - 13$"
      ],
      "text/plain": [
       "sqrt(x) - 13"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "3 + sy.sqrt(x) - 4 ** 2  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "uuid": "025f5084-4efc-4416-a564-b8a96ec426fd"
   },
   "outputs": [],
   "source": [
    "f = x ** 2 + 3 + 0.5 * x ** 2 + 3 / 2  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "uuid": "f427581e-50ba-45b9-a145-ff734be26e59"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.5 x^{2} + 4.5$"
      ],
      "text/plain": [
       "1.5*x**2 + 4.5"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.simplify(f)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "uuid": "59837e7c-a82f-44e5-8b87-b0fba310ffc5"
   },
   "outputs": [],
   "source": [
    "sy.init_printing(pretty_print=False, use_unicode=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "uuid": "ea1bbb85-a431-4e6c-a264-6ffb6df9f59e"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     2      \n",
      "1.5*x  + 4.5\n"
     ]
    }
   ],
   "source": [
    "print(sy.pretty(f))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "uuid": "e8702014-5347-4eac-b10c-39c5a54459c9"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  ___      \n",
      "\\/ x  + 0.5\n"
     ]
    }
   ],
   "source": [
    "print(sy.pretty(sy.sqrt(x) + 0.5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 136 ms, sys: 2.88 ms, total: 139 ms\n",
      "Wall time: 139 ms\n"
     ]
    }
   ],
   "source": [
    "%time pi_str = str(sy.N(sy.pi, 400000))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "uuid": "b8cf7cf4-56e2-4c6b-a70b-6592ec4bd6a6"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'3.1415926535897932384626433832795028841971'"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi_str[:42]  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "uuid": "595cc15e-99b8-43c7-ad89-db4500fdb860"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'8245672736856312185020980470362464176199'"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi_str[-40:]  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "uuid": "cc4b85c4-e603-4d36-9553-f3999e877fca"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 78 µs, sys: 1e+03 ns, total: 79 µs\n",
      "Wall time: 79.9 µs\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "80847"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time pi_str.find('061072')  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "uuid": "c310eeab-ac04-4015-a327-29628f9a9a11"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1, 1]"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.solve(x ** 2 - 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "uuid": "60076f9d-93f3-47e0-b5b6-87d1bca3d30a"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-2, 2]"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.solve(x ** 2 - 1 - 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "uuid": "6c9e1060-61e6-4b18-9be4-cb71b83fc5a7"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.858094329496553, -0.679047164748276 - 0.839206763026694*I, -0.679047164748276 + 0.839206763026694*I]"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.solve(x ** 3 + 0.5 * x ** 2 - 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "uuid": "fcd70615-0db5-4c5c-8246-245ba3c8501c"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[{x: -I*y}, {x: I*y}]"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.solve(x ** 2 + y ** 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integration and Differentiation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "uuid": "503329d1-812a-461c-9019-e6bb2a44125e"
   },
   "outputs": [],
   "source": [
    "a, b = sy.symbols('a b')  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [],
   "source": [
    "I = sy.Integral(sy.sin(x) + 0.5 * x, (x, a, b))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "uuid": "af7bf980-d65b-4567-8576-404e79c5ac65"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  b                    \n",
      "  /                    \n",
      " |                     \n",
      " |  (0.5*x + sin(x)) dx\n",
      " |                     \n",
      "/                      \n",
      "a                      \n"
     ]
    }
   ],
   "source": [
    "print(sy.pretty(I))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "uuid": "89d4c980-01d7-423a-a2ab-a9698787019a"
   },
   "outputs": [],
   "source": [
    "int_func = sy.integrate(sy.sin(x) + 0.5 * x, x)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "uuid": "38141163-7d29-4bd6-bfaf-9f9cfad86967"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "      2         \n",
      "0.25*x  - cos(x)\n"
     ]
    }
   ],
   "source": [
    "print(sy.pretty(int_func))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "uuid": "51d4481a-c6b6-4caa-b4d1-aa6fb04d4493"
   },
   "outputs": [],
   "source": [
    "Fb = int_func.subs(x, 9.5).evalf()  \n",
    "Fa = int_func.subs(x, 0.5).evalf()  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "uuid": "0c46e097-c665-440e-9d05-03c0d6e6d8d8"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "24.3747547180867"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fb - Fa  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [],
   "source": [
    "int_func_limits = sy.integrate(sy.sin(x) + 0.5 * x, (x, a, b))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "uuid": "657b13dc-101e-4c0c-9d15-43dbd418c3c8"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "        2         2                  \n",
      "- 0.25*a  + 0.25*b  + cos(a) - cos(b)\n"
     ]
    }
   ],
   "source": [
    "print(sy.pretty(int_func_limits))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "uuid": "5c65cea8-1e15-4efa-aba4-52888becaf19"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "24.3747547180868"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int_func_limits.subs({a : 0.5, b : 9.5}).evalf()  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "uuid": "b13767c8-4878-42c1-b738-0727b824b329"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "24.3747547180867"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.integrate(sy.sin(x) + 0.5 * x, (x, 0.5, 9.5))  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Differentiation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "uuid": "6ea93bb3-eb7c-4837-8327-621f3752b657"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5*x + sin(x)"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int_func.diff()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "uuid": "164131fd-a197-40ce-85d6-94221e808073"
   },
   "outputs": [],
   "source": [
    "f = (sy.sin(x) + 0.05 * x ** 2\n",
    "   + sy.sin(y) + 0.05 * y ** 2)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "uuid": "5d1ce51b-2536-469b-b516-56db1e86519a"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1*x + cos(x)"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "del_x = sy.diff(f, x)  \n",
    "del_x  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "uuid": "8d778c21-ed6a-43f0-8347-b95af334972a"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1*y + cos(y)"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "del_y = sy.diff(f, y)  \n",
    "del_y  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {
    "uuid": "588daaba-266a-43da-a4b9-baeb56fc1ac8"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.42755177876459"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xo = sy.nsolve(del_x, -1.5)  \n",
    "xo  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {
    "uuid": "2ef9beca-58f2-470b-a50e-6292bffbf2e0"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.42755177876459"
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yo = sy.nsolve(del_y, -1.5)  \n",
    "yo  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {
    "uuid": "c0812b29-c69f-4a30-aaf1-1447d1434cf4"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.77572565314742"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.subs({x : xo, y : yo}).evalf()  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {
    "uuid": "fd89d904-eab3-4035-941a-785c6a3f36cb"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.74632928225285"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xo = sy.nsolve(del_x, 1.5)\n",
    "xo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {
    "uuid": "cc99d8f4-e062-4c7e-85ae-46e96d3c8f19"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.74632928225285"
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yo = sy.nsolve(del_y, 1.5)  \n",
    "yo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {
    "uuid": "54622abe-cef1-47ff-bf23-b883cf588a91"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.27423381055640"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.subs({x : xo, y : yo}).evalf()  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://hilpisch.com/tpq_logo.png\" alt=\"The Python Quants\" width=\"35%\" align=\"right\" border=\"0\"><br>\n",
    "\n",
    "<a href=\"http://tpq.io\" target=\"_blank\">http://tpq.io</a> | <a href=\"http://twitter.com/dyjh\" target=\"_blank\">@dyjh</a> | <a href=\"mailto:training@tpq.io\">training@tpq.io</a>"
   ]
  }
 ],
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